## Probability and economics

30 Mar, 2017 at 16:02 | Posted in Economics | 2 Comments

Modern mainstream (neoclassical) economics relies to a large degree on the notion of probability.

To at all be amenable to applied economic analysis, economic observations allegedly have to be conceived as random events that are analyzable within a probabilistic framework.

But is it really necessary to model the economic system as a system where randomness can only be analyzed and understood when based on an a priori notion of probability?

When attempting to convince us of the necessity of founding empirical economic analysis on probability models,  neoclassical economics actually forces us to (implicitly) interpret events as random variables generated by an underlying probability density function.

This is at odds with reality. Randomness obviously is a fact of the real world. Probability, on the other hand, attaches (if at all) to the world via intellectually constructed models, and a fortiori is only a fact of a probability generating (nomological) machine or a well constructed experimental arrangement or ‘chance set-up.’

Just as there is no such thing as a ‘free lunch,’ there is no such thing as a ‘free probability.’

To be able at all to talk about probabilities, you have to specify a model. If there is no chance set-up or model that generates the probabilistic outcomes or events – in statistics one refers to any process where you observe or measure as an experiment (rolling a die) and the results obtained as the outcomes or events (number of points rolled with the die, being e. g. 3 or 5) of the experiment – there strictly seen is no event at all.

Probability is a relational element. It always must come with a specification of the model from which it is calculated. And then to be of any empirical scientific value it has to be shown to coincide with (or at least converge to) real data generating processes or structures – something seldom or never done.

And this is the basic problem with economic data. If you have a fair roulette-wheel, you can arguably specify probabilities and probability density distributions. But how do you conceive of the analogous nomological machines for prices, gross domestic product, income distribution etc? Only by a leap of faith. And that does not suffice. You have to come up with some really good arguments if you want to persuade people into believing in the existence of socio-economic structures that generate data with characteristics conceivable as stochastic events portrayed by probabilistic density distributions.

We simply have to admit that the socio-economic states of nature that we talk of in most social sciences – and certainly in economics – are not amenable to analyze as probabilities, simply because in the real world open systems there are no probabilities to be had!

The processes that generate socio-economic data in the real world cannot just be assumed to always be adequately captured by a probability measure. And, so, it cannot be maintained that it even should be mandatory to treat observations and data – whether cross-section, time series or panel data – as events generated by some probability model. The important activities of most economic agents do not usually include throwing dice or spinning roulette-wheels. Data generating processes – at least outside of nomological machines like dice and roulette-wheels – are not self-evidently best modeled with probability measures.

If we agree on this, we also have to admit that much of modern neoclassical economics lacks sound foundations.

When economists and econometricians – uncritically and without arguments — simply assume that one can apply probability distributions from statistical theory on their own area of research, they are really skating on thin ice.

Mathematics (by which I shall mean pure mathematics) has no grip on the real world ; if probability is to deal with the real world it must contain elements outside mathematics ; the meaning of ‘ probability ‘ must relate to the real world, and there must be one or more ‘primitive’ propositions about the real world, from which we can then proceed deductively (i.e. mathematically). We will suppose (as we may by lumping several primitive propositions together) that there is just one primitive proposition, the ‘probability axiom’, and we will call it A for short. Although it has got to be true, A is by the nature of the case incapable of deductive proof, for the sufficient reason that it is about the real world  …

We will begin with the … school which I will call philosophical. This attacks directly the ‘real’ probability problem; what are the axiom A and the meaning of ‘probability’ to be, and how can we justify A? It will be instructive to consider the attempt called the ‘frequency theory’. It is natural to believe that if (with the natural reservations) an act like throwing a die is repeated n times the proportion of 6’s will, with certainty, tend to a limit, p say, as n goes to infinity … If we take this proposition as ‘A’ we can at least settle off-hand the other problem, of the meaning of probability; we define its measure for the event in question to be the number p. But for the rest this A takes us nowhere. Suppose we throw 1000 times and wish to know what to expect. Is 1000 large enough for the convergence to have got under way, and how far? A does not say. We have, then, to add to it something about the rate of convergence. Now an A cannot assert a certainty about a particular number n of throws, such as ‘the proportion of 6’s will certainly be within p +- e for large enough n (the largeness depending on e)’. It can only say ‘the proportion will lie between p +- e with at least such and such probability (depending on e and n*) whenever n>n*’. The vicious circle is apparent. We have not merely failed to justify a workable A; we have failed even to state one which would work if its truth were granted. It is generally agreed that the frequency theory won’t work. But whatever the theory it is clear that the vicious circle is very deep-seated: certainty being impossible, whatever A is made to state can only be in terms of ‘probability ‘.

John Edensor Littlewood

This importantly also means that if you cannot show that data satisfies all the conditions of the probabilistic nomological machine, then the statistical inferences made in mainstream economics lack sound foundations!

1. In theory I am not a Bayesian, but I would admit that as a pedestrian I generally act as if I am one. But, being British, I found that the first time I went abroad I had to radically revise my probability estimates (i.e., not using Bayes’ rule), and similarly when walking in a mixed-usage area with silent electric vehicles.

Similarly it seems to me that mainstream economic theory is quite reasonable about what normally happens – but sometimes the abnormal happens.

One problem with Bayes’ rule is that it supposes that there is some fixed context that the probability estimate apply to, whereas sometimes you have got the context wrong, and may need to go back and re-assess the old evidence.

Sometimes it is appropriate to apply habitual thinking, but also to think about possible abnormalities. Probability theory is still useful, just not in the way mainstream economists use it.

2. There is no “probabilistic nomological machine” regarding the behavior of traffic along a busy road. So we can never know with certainty whether it will be safe to cross a busy road on foot. As Prof. Syll frequently and correctly insists, quoting Keynes, “We simply do not know”.
Does this mean we should stand forever at the kerb dithering in a state of philosophical “fundamental” uncertainty? Of course not. In an uncertain world we have to disregard such philosophical nihilism. We have to form judgments and take risks even with imperfect information.
Every day billions of people cross roads safely. There is indeed no “free lunch” which enables them to do this. They have to learn from the experiences of others and themselves to form a mental model of the about the likelihoods and risks involved. Moreover any such model and its application is bound to be imperfect, as witnessed by accident statistics.
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The mental models used every day by pedestrians are not rigorously formulated as mathematical probability models, but it makes good sense to regard them as having similar properties. As in Haavelmo’s probability model, at a given type of crossing location there is a very large statistical “population”, namely combinations of different circumstances and resultant traffic behaviours which could arise. Unfortunately, we have knowledge of only a limited sample of situations and outcomes in the past. Moreover, the best specification of any model is always open to question.
Even so such models have obvious practical use. Indeed they are vital in everyday life, despite inevitable imperfections, the risks of using them and the lack of any philosophical “probabilistic nomological machine”.
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Ditto for cycling, driving a car, nutrition, health, and likewise economics.

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